In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
A similar set of polynomials, based on a similar generating function, is the family of Euler polynomials.
The Bernoulli polynomials Bn admit a variety of different representations. Which among them should be taken to be the definition may depend on one's purposes.
The generating function for the Bernoulli polynomials is
The generating function for the Euler polynomials is
Representation by a differential operator
The Bernoulli polynomials are also given by
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. integrals below. By the same token, the Euler polynomials are given by
Another explicit formula
An explicit formula for the Bernoulli polynomials is given by
Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has
where ζ(s, q) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.
The inner sum may be understood to be the nth forward difference of xm; that is,
where Δ is the forward difference operator. Thus, one may write
This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
where D is differentiation with respect to x, we have, from the Mercator series
As long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m.
An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
The above follows analogously, using the fact that
Sums of pth powers
(assuming 00 = 1). See Faulhaber's formula for more on this.
The Bernoulli and Euler numbers
The Bernoulli numbers are given by
This definition gives for .
An alternate convention defines the Bernoulli numbers as
The two conventions differ only for since .
The Euler numbers are given by
Explicit expressions for low degrees
The first few Bernoulli polynomials are:
The first few Euler polynomials are:
Maximum and minimum
At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,
which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys
unless n is 2 modulo 4, in which case
(where is the Riemann zeta function), while the minimum obeys
unless n is 0 modulo 4, in which case
These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.
Differences and derivatives
The Bernoulli and Euler polynomials obey many relations from umbral calculus:
(Δ is the forward difference operator). Also,
Zhi-Wei Sun and Hao Pan established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then
Note the simple large n limit to suitably scaled trigonometric functions.
This is a special case of the analogous form for the Hurwitz zeta function
This expansion is valid only for 0 ≤ x ≤ 1 when n ≥ 2 and is valid for 0 < x < 1 when n = 1.
The Fourier series of the Euler polynomials may also be calculated. Defining the functions
for , the Euler polynomial has the Fourier series
Note that the and are odd and even, respectively:
They are related to the Legendre chi function as
The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.
Specifically, evidently from the above section on integral operators, it follows that
Relation to falling factorial
The Bernoulli polynomials may be expanded in terms of the falling factorial as
denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
denotes the Stirling number of the first kind.
For a natural number m≥1,
Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:
Periodic Bernoulli polynomials
A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.
Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions.
The following properties are of interest, valid for all :
- D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", American Mathematical Monthly, volume 47, pages 533–538 (1940)
- Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica. 125: 21–39. arXiv:math/0409035. Bibcode:2006AcAri.125...21S. doi:10.4064/aa125-1-3.
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- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 (See chapter 12.11)
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- Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0. (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
- Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 0-521-84903-9.